The Donnan potential revealed

Selective transport of solutes across a membrane is critical for many biological, water treatment and energy conversion and storage systems. When a charged membrane is equilibrated with an electrolyte, an unequal distribution of ions arises between phases, generating the so-called Donnan electrical potential at the solution/membrane interface. The Donnan potential results in the partial exclusion of co-ion, providing the basis of permselectivity. Although there are well-established ways to indirectly estimate the Donnan potential, it has been widely reported that it cannot be measured directly. Here we report the first direct measurement of the Donnan potential of an ion exchange membrane equilibrated with salt solutions. Our results highlight the dependence of the Donnan potential on external salt concentration and counter-ion valence, and show a reasonable agreement with current theoretical models of IEMs, which incorporate ion activity coefficients. By directly measuring the Donnan potential, we eliminate ambiguities that arise from limitations inherent in current models.


Supplementary Tables
. This value refers to the equilibrium water content of Na + counter-ion form membranes equilibrated in ultrapure DI water. It represents the water uptake in only the active ion exchange polymer phase in the composite CR-61 membrane (i.e., the contribution of the fabric backing has been removed as detailed elsewhere 2 ).

Supplementary Notes
Supplementary Note 1.

Attenuation of photoelectron peaks and estimating the solution layer thickness
The main challenge of the membrane/liquid study with APXPS is related to the short inelastic mean free paths (IMFPs) of photoelectrons, the very same characteristic that makes the technique surface sensitive. In our experimental setup, membrane related core level photoelectrons (i.e., S where %& ! " ' is the number density of sulfonate groups, %& ! " ' is the photoionization cross-section of oxygen, and 'C'D and +,-./0,1 are the inelastic mean free paths (IMFP) in membrane and solution, respectively. Unlike the case for common solid samples, the liquid phase water (LPW) O 1s signal intensity in our experimental system arises from both the liquid layer and the sorbed water inside the membrane. The intensity of the O 1s peak for LPW is: where 2 ' and 2 3 are the number density of oxygen atoms and 2 ' and 2 3 are the photoionization cross-section of sorbed water inside the membrane and oxygen in the solution layer, respectively.
Water content inside the solution-equilibrated membrane is estimated from the number of sorbed water molecules per sulfonate group for CR-61 membranes exposed to water vapor (hydrated conditions) before the dip and pull procedure. In order to do that, the ratio of O 1s peak areas of LPW to sulfonate components is multiplied by a stoichiometric factor of 3, to be representative of With an estimated liquid layer ~17-21 nm thickness, the intensity of the O 1s peak for the SO = > component decreases by ~88% of its initial (dry) value due to attenuation through the liquid layer, which requires more collection time to get sufficient signal to noise of the spectra.

Numerical Simulation of Potential Profile at Membrane/Solution Interface
To simulate the electrical potential distribution across a charged membrane in equilibrium with an electrolyte solution, we used a model previously introduced by Ohshima et al. 8 . It is assumed that our ion permeable membrane has uniformly distributed, fixed negative charges, and the membrane is in equilibrium with a large volume of a symmetrical electrolyte solution of concentration 4 + and valence . We select the x-axis to be the direction normal to the membrane surface, so the plane at x = 0 coincides with the membrane/solution interface (see Supplementary Fig. 4).
We assume that the electrical potential at position x in the solution and membrane regions satisfies the Poisson-Boltzmann equation, as follows: For the solution side, where x<0: For the membrane side, where x>0: Boundary conditions for our experimental system are: 1) The electrical potential of the bulk solution is equal to zero: As gathered from Supplementary Fig.3., y(x) exhibits a Gouy-Chapman type diffuse layer decay for x < 0.

Simulation XP Spectra from the Potential Distribution at the Membrane/Solution Interface
After numerically solving the related Poisson-Boltzmann equations to simulate the potential distribution at the membrane/solution interface, we used Python to simulate the S 1s and O 1s core level XPS spectra. The liquid phase water (LPW) and membrane phase related-core levels (S 1s) spectra are described as the convolution of the binding energy shifted spectra of the elements as a function of their position within the potential drop with respect to the membrane/solution interface. 6 The starting spectra were modelled as a Gaussian function. Three pieces of information are needed to numerically simulate the starting spectra: i) the binding energy (BE), which determines the center of each individual peak; ii) the Gaussian broadening, which is obtained from the experimental full width at half maximum (FWHM); and iii) peak area, which is determined from the number density of each element, integrated over the exponential escape probability of the photoelectron intensity (Beer-Lambert Law). 11,12 Binding energy differences between various peaks and the related FWHM values were obtained from our previous study of the same polymer. 3 Number densities for each element used in the simulations are given in Supplementary Table 4.
S 1s and O 1s core level XPS peaks were obtained by summation of individual spectra generated every 0.1 Å steps within the potential drop. The binding energy location of each individual spectrum shifted 1 V:1 eV, following the electrical potential profile, as a function of distance from the interface. In addition, the exponential decrease in intensity for each individual spectrum as a function of depth is modelled by the Beer-Lambert law: where is the probing depth of the membrane with respect to the interface, and is the inelastic-mean-free-path (IMFP) of the photoelectron. The Tanuma

Binding energy shifts of other core levels
In theory, the Donnan potential of CR-61 membranes equilibrated with aqueous salt solutions can also be assessed from the binding energy shifts of other membrane related core levels (i.e., O 1s-SO3 and C 1s). Supplementary Table 6 shows the binding energies of fitted XP spectra for O 1s-LPW, O 1s-GPW, O 1s-SO3, S 1s-SO3 and Na/Mg 1s. C1s core level spectra are also collected during the experiments. However, the C1s region consists of at least 5 different chemistries including aromatic, aliphatic, and C-SO3 from the membrane. Additionally, the C 1s region is susceptible to further changes upon exposure to water (in the form of adventitious carbon and other carbonaceous species, i.e., C-O and C=O). Deconvolution of this peak is challenging since the many individual contributions forming the C 1s peak are not well resolved. We also could not constrain the area of any membrane components during the fitting procedure, due to the unknown percentage of cross linker in the membrane assembly. This made the peak-fitting process challenging and following such small binding energy differences in the region unreliable. That is why we have not presented the binding energies of C 1s chemistries here. Representative C 1s spectra for each concentration NaCl and MgCl2 are provided in Supplementary Fig. 7 with the binding energy of possible individual chemical carbon contributions. As can be gathered from the figure, no carbonate species are detected in the C 1s spectra, which generally appears at binding energies higher than 288.5 eV. 6 This is important especially for this type of experiment as competitive ion sorption by other ions in solution (e.g., carbonates from CO2 speciation) could introduce error in the Donnan Potential measurement.
As predicted, the binding energy shifts in membrane O 1s are consistent with those in S 1s (see Supplementary Fig. 8 and Table 6). However, the O 1s region requires multi-peak fitting, which introduces additional uncertainties and increases the error bars of the measurement. In addition, due to the nature of this experimental technique, membrane related core level photoelectrons are attenuated through both the liquid and the gas phase. Low binding energy peaks shows relatively low photoelectron intensities due to their low cross sections. Considering these factors, in the main text, we chose a single component, high binding energy S 1s core level to measure the Donnan potential, leading to smaller error bars and more precise measurements.
We would like to note that the counter-ion binding energies show a different behavior. Contrary to our observations in the immobile membrane peaks, no clear concentration dependence is observed in the counter-ion related binding energy of NaCl-and MgCl2-equilibrated CR-61 (see Supplementary Fig. 9 and 10). This behavior may arise from the weak interaction with the membrane sulfonate charges and/or the strong hydration shell of counter-ions. We are currently devoting time and effort to better understand this phenomenon, and it will be the focus of a future manuscript.
In addition to counter-ions, co-ion (Cl 1s) core level spectra were also collected during the experiments, but no peak was observed for electrolyte concentrations lower than 1 M. Considering the nature of ion exchange membranes, where co-ion concentration inside the membrane is very low compare to the counter-ion concentration, which is dictated by the number of fixed charges (i.e., for CR-61 = 3.2 M), we believe that observed counter-ion specific peaks were coming mainly from the ions inside the membrane at lower salt concentration. This sorption behavior is supported by quantitative analysis and numerical simulations on Na 1s/S 1s and Mg 1s/S 1s peak area ratios given in Supplementary Note 6. As predicted, peak area ratios of counterion to membrane related core levels do not show any obvious trend or change at lower concentration. On the other hand, at 0.3 M salt concentration, Na 1s/S 1s and Mg 1s/S 1s area ratios starts to increase due to the additional contribution from detection of counter-ions in the liquid layer.
Since the detected Mg 1s and Na 1s regions at 1M solution concentration are a convolution of peaks coming from ions in both the membrane and liquid phase, the binding energy values at 1 M external solution concentration are excluded from the plot given in Supplementary Fig. 10.
Unfortunately, it was not possible to deconvolute the solution phase ions from the ions inside the membrane because of small binding energy/electrical potential differences between them, which were not enough to form any peak separation or broadening. However, the small drop in overall Mg 1s BE at 1 M solution concentration (Supplementary Table 6. and Fig. 9b) may arise from an additional contribution from the liquid layer which was not detectable at lower concentrations.
In addition, no significant trend for binding energies of O 1s-GPW peak was observed with respect to the external solution concentration.

Numerical simulation of the photoelectron intensity
Since instrumental parameters which depend on the photoionization cross-section of elements at a given X-Ray energy, X-ray flux at a given X-ray energy, the orbital specific asymmetry, and the spectrometer efficiency for a given kinetic energy (KE). S must be taken into an account when quantifying photoelectron peaks with different KE.
Similarly, the intensity of the S 1s peaks of a CR-61 membrane underneath a liquid layer of thickness d, over the entire concentration range probed, is obtained from: By substituting the values of IMFPs listed in Supplementary Table 9, the relative intensities of the Na 1s or Mg 1s and S 1s XPS peaks are simulated over the entire salt concentration range probed. (Supplementary Fig. 11) The simulated intensity ratios agree with experimental observations. As expected, peak area ratios of counterion to membrane related core levels do not show any obvious trend or change at lower concentration. On the other hand, Na 1s/S 1s and Mg 1s/S 1s area ratios start to increase at high concentrations due to additional contribution from detection of counter-ions in the liquid layer. It needs to be highlighted that these intensity simulations were performed using homogeneous and well-defined layered structures and interfaces as an approximation of the real configuration, where most likely concentration gradients and mixed regions exist. In addition, it was previously established that the surface composition of the salt solutions are enhanced in the halide anion concentration (and thus attenuation of the cation) compared with the bulk of the solution at the liquid/vapor interface. 14,15 Given these assumptions there is reasonable agreement between our experimental and simulated data, which reveal a similar trend of the primary detectable spectra contributions.

Experimental uncertainties and statistical significance
Since our approach to directly measuring the Donnan potential at the membrane/solution interface is based on following small binding energy shifts of membrane related core level XP spectra, we utilized a number of statistical methods to estimate the experimental uncertainties, determining outlier data points and evaluating statistical significance of our data. As mentioned in the main text, the error bars used to represent experimental uncertainty are determined from the standard deviation of repeated measurements of 4-5 different positions in each case.
In addition, we employed the statistical "Q test" to determine if a single data value is an outlier in a distinct sample size. This test essentially calculates the ratio between the putative outlier's distance from its nearest neighbor and the range of values. Upon comparing the calculated Q to the theoretical Q, one very large (2477.12 eV S 1s binding energy for CR-61 equilibrated with 0.1M NaCl) and one very small (2476.93 eV S 1s binding energy for CR-61 equilibrated with 0.1M NaCl) value were rejected from the data set by the Q-test with 95% confidence.
To demonstrate that the slopes of linear fit to our experimentally measured binding energies of S 1s peak are statistically different from zero and from each other for NaCl and MgCl2 equilibrated CR-61 membranes, we ran a statistical t-test (inferential statistical test used to determine if there is a significant difference between the means of two groups) and an ANOVA (Analysis of Variance) analysis using Origin software. Further information regarding these statistical analyses can be found elsewhere 16 . The data in Fig. 3 are presented using the raw data in Supplementary Fig. 14, and the related parameters from the statistical analysis are given in Supplementary Tables 10 and   11 for NaCl and MgCl2, respectively. Since the t and F values exceed the critical values of t 16 and F 16 at 95% confidence level for specific degrees of freedom (DF), we conclude that the slopes are statistically different from zero and from each other.

Model Predictions of the Donnan Potential
As discussed in the main text, the presence of fixed charge groups inherently leads to an unequal distribution of counter-ions and co-ions in IEMs. This phenomenon generates an electrical potential at the membrane/solution interface, referred to as the Donnan potential 17 (Ψ !"##$# ), that acts to attract counter-ions into the membrane and restrict co-ions from entering the membrane 18 .
Following a standard thermodynamic treatment, Ψ !"##$# is defined as follows 17,18 : where is the universal gas constant, is the temperature, is where , , , , and refer to counter-ions, co-ions, the membrane, the solution, and the stoichiometric coefficient of an ion, respectively. Incorporating electroneutrality conditions into Donnan's model yields an equation that can be solved for ] ( , based on the fixed charge concentration of the membrane, < ( , and expressions for 0 in each phase. Electroneutrality in a charged membrane is expressed as 18,19 : where is the valence of the fixed charge groups (e.g., -1 for CR-61). In Donnan's original theory (the classic Donnan model), ion activity coefficients are eliminated from equation S11 by assuming either ion activity coefficients of unity (i.e., thermodynamic ideality) or equality of ion activity coefficients: < ( values were calculated from previously reported data for the ion exchange capacity (IEC) and water uptake ( . ) of CR-61. The following expression relates these parameters 19 : where ^ is the density of water. Values for NaCl-and MgCl2-equilibrated CR-61 were calculated using data from Kamcev et al. 19 20 Manning derived an expression for the excess free energy of a polyelectrolyte with added salt, accounting for the electrostatics that lead to the condensation of counter-ions near fixed charge groups. Manning's theory requires a single parameter, , which is defined as 20 : where * is the Bjerrum length, is the average distance between fixed charge groups, is the protonic charge, J is the vacuum permittivity, is the dielectric constant of the solution, is where is defined as < ( / + ( , and where + ( is the mobile salt concentration in the membrane and is stoichiometrically related to the co-ion concentration in the membrane: Equations S12, S13, and S17-S19 were solved numerically to predict 0 ( and 0 ( in CR-61 equilibrated with 0.001-1 M NaCl and MgCl2, using known < ( values and = 1.83 19 . Prior work has shown good agreement between these predictions and experimental ion sorption data from CR-61 19,2 . The Pitzer model was used to calculate the mean ion activity coefficient, ± + , in the external solution, due to the model's excellent agreement with ± + data over a broad range of ionic strength 24 . ± + is defined as the geometric mean of the individual ion activity coefficients 22 : ± + = †( S + ) \ 1 ( > + ) \ " 2 1 1 2 " (S20) Using the predicted values of 4 ( and 4 ( from the approach outlined above, theoretical Ψ !"##$# values were calculated via equation S8. Because the Pitzer model is expressed in terms of ± + , values of 4 + for NaCl and MgCl2 at a given ionic strength were approximated using ± + values for KCl from the Pitzer model at the same ionic strength. Under the assumption that 0 + for K + and Clare equal, the activity coefficient of Clat a given ionic strength is known and can be used to estimate the activity coefficients of Na + and Mg 2+ in NaCl and MgCl2 at the same ionic strength (Eq. S20). Although crude, this estimation is consistent with some aqueous electrolyte data and studies in the literature. 22 Predicted values, using the Manning/Donnan model, of 4 ( and 4 ( at a given external NaCl and MgCl2 salt concentration are presented in Supplementary Table 14 and Supplementary Table   15, respectively.

Comparison between experimental data and classical Donnan model
Supplementary Fig. 15a and 15b  This phenomenon results in highly non-ideal thermodynamic behavior, especially at low electrolyte concentrations. In contrast, when an IEM is equilibrated with a highly concentrated solution, the fixed charge groups are electrostatically screened by sorbed salt, and the thermodynamic environments of the solution and membrane phases become more similar. In this way, the ion activity coefficients in the membrane and in the external electrolyte solution also become closer to one another, such that the classic Donnan model can provide a reasonable prediction of the Donnan potential in CR-61 equilibrated with 1 M NaCl.
For MgCl2 equilibrated CR-61 (Supplementary Fig. 15b) have shown that activity coefficients in the membrane are more similar to activity coefficients in the external solution for CaCl2 equilibrated CR-61 than for NaCl equilibrated CR-61, particularly at higher external salt concentrations. 25 The ability of the classic Donnan model to reasonably predict the Donnan potential, within the error of our measurements, arises from the similar behavior of MgCl2 equilibrated to that of CaCl2.